Digital imaging has been used to measure the deformation of deformable material specimens and structures. These displacement measurement methods have gained significant attention the last two decades, because of the great impact of digital imaging evolution. Modern digital cameras provide a cost effective and highly reliable tool for recording and processing images of an experiment using a personal computer. Experimental mechanics have greatly benefited from those capabilities, and various methods have been developed for the determination of displacement and strain fields.
Both pure grid methods and digital image correlation methods have been proposed for providing full-field measurements of displacement and strains.
In pure grid methods, a uniform grid is applied to the surface of a specimen, and the measurement of deformation relies on the motion of the grid. Many of these methods rely on specialized means for application of a uniform grid. It can be difficult to apply a uniform grid to irregularly shaped bodies, and any inaccuracies in the application of the grid are a major source of errors in the measurement of deformation.
Pure grid methods are described in Sevenhuijsen, P. J., “Two simple methods for deformation demonstration and measurement”, Strain, Vol. 17, pp. 20-24 (1981); Parks, V. J., “Strain measurements using grids”, Opt. Eng., Vol. 21, pp. 633-639 (1982); Sevenhuijsen, P. J., “Photonics for deformations”, Proc 5th Int. Congr. On Expt. Mechanics, SESA, Montreal, (June 1984); and Sevenhuijsen, P. J., “The Photonical, Pure Grid Method”, Optics and Lasers in Engineering, Vol. 18, pp. 173-194, (1993).
Digital image correlation methods are described in Peters, W. H., Ranson, W. F., “Digital imaging techniques in experimental stress analysis”, Opt. Eng. Vol. 21, pp. 427-432, (1982); Bruck, H. A., McNeil, S. R., Sutton, M. A., and Peters W. H., “Digital image correlation using Newton-Raphson method of partial differential correction”, Expt. Mech. Vol. 28, pp. 261-267 (1989); and Cheng, P., Sutton, M. A., Schreier, H. W., McNeill, S. R., “Full-field speckle pattern image correlation with B-Spline deformation function”, Expt. Mech., Vol. 42, pp. 344-352, (2002).
The performance of methods based on digital image correlation, which rely on an applied speckle pattern, can be highly sensitive to the application method and specimen surface. Schreier, H. W. Sutton, M. A., “Systematic errors in digital image correlation due to undermatched subset shape functions”, Expt. Mech., Vol. 42, pp. 303-310, (2002) discusses the sensitivity of the method to very specific qualitative and quantitative characteristics of the speckle pattern.
Additional grid-based methods are described in Sirkis, J. S., “System response to automated grid methods”, Opt. Eng., Vol. 29, 1485-93, (1990) and Andersen, K., Helsch, R., “Calculation of grating coordinates using correlation filter techniques”, Optik, Vol. 80, pp. 76-79, (1988). U.S. Pat. No. 7,377,181 to Christ, Jr., et al. discloses the use of coded marks.
Bremand, F. and Lagarde, A., “Two methods of large and small strain measurement on a small size area”, Proc. SEM Spring Conf. On Expt. Mechanics, Keystone, Colo., USA, pp. 173-176, (1986) discloses a method of applying a Fourier transform of the grid pattern.
Mesh-free methods are described in Andrianopoulos, N. P., “Full-field displacement measurement of a speckle grid by using a mesh-free deformation function”, Strain, Vol. 42, 265-271, (2006), in Andrianopoulos, N. P. and Iliopoulos, A. P. “Displacements Measurement in Irregularly Bounded Plates Using Mesh Free Methods”, 16th European Conference of Fracture, Alexandroupolis, Greece, Jul. 3-7, 2006.
Two dimensional random-grid mesh-free techniques are disclosed in Andrianopoulos, N. P. and Iliopoulos, A. P., “Strain measurements by a hybrid experimental-numerical method using a mesh-free field function”, Honorary Volume for Professor P. S. Theocaris, Armenian Academy of Sciences, 31-41, (2005) and in Iliopoulos, A. P., Andrianopoulos, N. P., “An Approach to Analyze Errors Introduced in the Random Grid Strain Measurement Method”, Strain, Vol. 46, pp. 258-266, June 2010 (published online November 2008), and in copending patent application Ser. No. 12/793,594 to Michopoulos et al., published as U.S. Patent Publication No. 20100310128, the entire disclosure of which is incorporated herein by reference.
Three dimensional random-grid mesh-free techniques that exploit differentiations of the displacement fields are disclosed in copending patent application Ser. No. 13/565,698 to Michopoulos, et al., published as U.S. Patent Publication No. 20130063570, the entire disclosure of which is incorporated herein by reference.
Calculation of strain by differentiation of displacement fields is a typical practice in full field measurement methods. In addition, there are techniques, such as Shearography and Moiré interferometry that exploit implicit differentiations of the displacement fields, but compute only a subset of the strain components, require specialized instrumentation, are impractical for non-planar cases, and are sensitive to out of plane motions.
Strain field estimation based on differentiation of displacement fields utilized by these techniques implies that the strain compatibility equations remain satisfied for the entire domain of observation over the entire loading history, such as the continuum hypothesis remains valid. However, when surface discontinuities or strain localization occur due to damage initiation, the continuum hypothesis is no longer valid throughout the entire field of observation. Therefore, full field measurement results based on the anticipated validity of the continuous hypothesis yield both inaccurate and indeterminate measurements, thus leading to false qualitative and quantitative conclusions.
Experience with the utilization of high throughput multi-degree of freedom automated mechatronic testing machines (i.e., which are capable of loading specimens multiaxially in conjunction with energy-based inverse constitutive characterization methodologies for composite materials) has underlined the need for both automated full field data processing and an accurate encapsulation of deformation fields. Such concerns become increasingly more important in regions of a specimen that are close to, or have undergone failure and accordingly the medium can no longer be considered as a continuum. In these areas, the micro-cracks and cracks break the validity of the continuum hypothesis and the associated strain compatibility relations.
Another important issue arising from the derivation of strain from a full field displacement representation is that of relative lower accuracy near and on the boundaries of the implemented analytical representation. As the strain components are defined in terms of the derivatives of the displacement components, it is expected that the analytical approximation of the strain components will be somewhat problematic, as the displacement approximation does not impose spatially driven constraints on the formulation of its derivatives. Although it is possible to reformulate the Mesh-Free approximation of the displacement fields to take into consideration such information, this would be impractical and would require considerable and customized effort to manually guide the approximation from an algorithmic perspective in a manner that is aware of the boundaries and it remains general without need for customization from geometry to geometry.
A final issue of concern arises from the fact that if there is noise in the displacement fields, their differentiation will only amplify it for the evaluation of the strain fields.
Accordingly, there is a need for a system and method for calculating the full field of strain quantities from full field digital images of deforming specimens that addresses these and other issues. The numerical and analytical method of the present invention succeeds in approximating the strain tensor field directly from directional engineering strain quantities without requiring or enforcing the satisfaction of the compatibility conditions.
Since the present invention introduces Direct Strain Imaging (DST) as a new full field measurement method, the steps involved with the implementation of the traditional Mesh-less Random Grid (MRG) method will first be described. This will help delineate the differences and provide the basis of performance comparison between MRG and DSI methods.
Heretofore, a conventional experimental procedure for measuring a full field of deformation quantities (i.e., according to most full field methods in general, and MRG method in particular), can be outlined as follows:                1. A specimen is marked with an appropriate visible pattern that consists of a random distribution of dots distinguishable from the background.        2. If the experiment is to take into consideration out of plane motion, two or more cameras are used so that the deformation is stereoscopically reconstructed. The projective characteristics of the cameras are identified through an appropriate calibration procedure.        3. The specimen is placed in the mechanical testing machine and one image per camera is captured in the un-deformed configuration prior to the initiation of the loading sequence.        4. While the experiment is taking place, successive images of the deforming specimen are captured.        5. The images are processed and for each frame, the coordinates of appropriate points (nodes) are calculated. Those nodes may be for example geometric centroids of dots (for the case of grid methods), appropriate boundaries of geometric entities, correlated sub regions (for the case of DIC), or other such features, etc.        6. Using an interpolation or approximation scheme, a representation of the displacement field is obtained. In the MRG method, the field is represented by a continuous mesh-less approximation.        7. The strain at any point in the domain is obtained by differentiation of the displacement fields based on the definition of the strain measure as a function of the displacements.        
Noise introduced by various sources is the most dominant source of error, which plagues all conventional full field measurement methods. In its presence, the MRG method has been shown to perform very well compared to other methods. The main reason for its improved performance is that the approximation scheme of the mesh-less representation works as a filter that deals well with image acquisition noise. Unfortunately, since not all noise can be removed, there is still room for improving the accuracy levels of full-field techniques. The method proposed on the present approach mitigates most of the issues described in this section.